Instruments used to measure a particular parameter may be affected by the variation of other parameters. For example, measurement of the mass of material deposited on a microbalance may be adversely affected by a variation in temperature or pressure.
A microbalance, examples of which are described in U.S. Pat. Nos. 3,926,271 and 4,391,338, typically comprises an oscillating element mounted with one end fixed and the other end free. The free end typically has a filter (or other mass-receiving element) mounted thereto. The oscillating element may also be hollow, and in such a case a fluid is typically drawn through the filter and through the oscillating element, thereby to trap suspended particles within the fluid in or on the filter. The resulting increase in the mass of the filter decreases a resonant frequency of the oscillating element. The decrease in the resonant frequency of the oscillating element is related to the increase in mass of the filter, which in turn is equal to the mass of the suspended particles trapped in the filter. Because the oscillating element has the ability to continually indicate the mass of the suspended particles it is ideal for indicating the change in mass of the trapped or suspended particles in near real time or over a measured period of time.
As the temperature of the microbalance""s oscillating element changes, the resonant frequency of the oscillating element changes, even though the mass on the filter substrate secured to the oscillating element may remain unchanged. As the measured mass is based on the resonant frequency, an error is introduced in the mass determination. This temperature sensitivity results mainly from a change in the modulus of elasticity of the material, from which the oscillating element is made, as the temperature changes.
One way of addressing the concern of the temperature sensitivity of the microbalance, or other instrument, is to select a material of construction that has minimal sensitivity to changes in temperature. For a microbalance, great care can be applied to the formulation of the glass from which the oscillating element is made, thereby to attain the desired characteristics while attempting to optimize manufacturability and minimize the temperature sensitivity of the desired variables. In particular, one way of reducing the temperature sensitivity of the microbalance is to use a shaped oscillating element made of a glass having a low temperature coefficient of elastic modulus. In the end, compromises must be made at the expense of the accuracy, manufacturability and cost of the entire system.
According to one aspect of the invention, the temperature sensitivity of an instrument, for example the microbalance described above, is reduced by maintaining the instrument, or at least a temperature-sensitive element thereof, at a constant temperature. This is achieved by: applying heat to the instrument; measuring a parameter that is indicative of the temperature of the instrument; and controlling the amount of heat applied to the instrument to maintain the measured parameter substantially constant.
For example, this control may be accomplished by affixing a resistive heater on the oscillating element of the microbalance. The resistive heater can be wound onto the oscillating element, vacuum deposited, or applied by any other means. Some glass formulations allow embedding platinum heater windings directly within the glass. Radiant or other types of heating, such as convective and conductive, can also be employed. A radiant heater would be positioned appropriately next to or around the oscillating element to provide heat thereto. The parameter that is used to control the amount of heat supplied to the heater can be the resistance of the heating element (which is dependent on the temperature), or the output of an appropriately positioned temperature sensor.
A further error in the use of microbalances having hollow oscillating elements is caused by temperature and/or pressure changes in the fluid located within the cavity of the hollow oscillating element. As the temperature or pressure of the fluid within the cavity of the hollow oscillating element varies, so will the density of the fluid. Assuming that the interior volume of the cavity of the hollow oscillating element remains substantially constant, a variation in the density of the fluid will result in a variation of the mass of the fluid located in the cavity of the hollow oscillating element. That is, the effective mass of the oscillating element will vary with temperature and/or pressure changes in the fluid located therein. This variation in the effective mass of the oscillating element will in turn affect the resonant frequency of the oscillating element.
The pressure-dependent error frequently manifests itself as a perceived negative mass over time. As the filter element loads up with particulate or other forms of flow-impeding elements, the pressure within the cavity of the hollow oscillating element will decrease due to the increased resistance of the filter element. As the pressure of the fluid decreases, so does the density, reducing the mass of the column of fluid within the cavity of the hollow oscillating element. This in turn will increase the resonant frequency of the oscillating element, indicating a false reduction in the mass of the filter element and its entrapped matter.
Similarly, if the fluid gets colder, the density of the fluid column in the cavity of the hollow oscillating element will increase, increasing the mass within the cavity of the hollow oscillating element. The resonant frequency of the oscillating element will be correspondingly lower, thereby indicating an erroneously high mass. The reverse is true if the fluid temperature increases.
The compensation for the variation in temperature or pressure can be performed as follows, using an idealized gas as a fluid, with Boyle""s law (PV=nRT) to approximate the behavior of the gas. It will be appreciated that other models and equations can be used for performing the pressure/temperature/density compensation, and that other models and/or equations can be used to represent the dynamics of the instrument in question.
Referring to FIG. 1, the principle of operation of a microbalance 10 can be represented by m=k/f2, where m is the mass in grams, f is the frequency in Hz, and k is the spring constant in g*Hz2.
For a particular microbalance, the spring constant can be determined by using two values of m one for xe2x80x9czeroxe2x80x9d mass (i.e. the system mass only), and one for an additional mass added to the xe2x80x9czeroxe2x80x9d mass. The system mass is of course not actually zeroxe2x80x94we xe2x80x9ctarexe2x80x9d the system mass out for purposes of convenience, much like a post office scale is zeroed before the letters are placed in a box on the scale. This ensures that the mass of the letters only is considered, and not the mass of the box.
The equation for the spring constant can be derived as follows:k=(m1m0)/(1/f121/f02)Using exemplary values of m1=0.075g, m2=0, f1=250 Hz and f=311.314 Hz yields a value of k=13.200 g*Hz2, obtained at a temperature of 20xc2x0 C. and 29.92 inHg, with the fluid being an air mixture. At this temperature and pressure, the air mixture has a density of 1.200 g/l.
The actual system mass at this temperature can now be determined by substituting the determined value of k and the observed value of f for the xe2x80x9czeroxe2x80x9d mass condition. Using these two values, we arrive at a system mass of 0.136199 g. Part of this system mass results from the column of air in the cavity of the hollow oscillating element.
As mentioned, the mass at the xe2x80x9czeroxe2x80x9d condition (the system mass only) includes a mass component derived from the mass of the fluid column in the cavity of the hollow oscillating element at the particular fluid density. The effect of the fluid column on the system mass can be determined by measuring the xe2x80x9czeroxe2x80x9d or system mass at a different fluid density, as follows.
Repeating the test described above at a reduced air pressure of 20.00 inHg , the new density (from Boyle""s law) is rho=rho1P1/P * T/T1=0.8021 g/l. The observed frequency at 20 inHg is 311.4 Hz, which indicates a mass of m=k/f2=0.1361 24 g. This new system mass, and its associated change in frequency, have resulted from the decrease in the fluid density (and hence the mass of fluid in the cavity of the hollow oscillating element). The change in mass, delta_m=0.136199 0.136124=0.000075 g, which has occurred as a result of a change of density, delta_rho, of 1.20 g/l 0.8021 g/l=0.3979 g/l.
The xe2x80x9cactivexe2x80x9d volume, V, (i.e. the effective volume that contributes to the variation in the frequency) can be determined as follows:V=delta_m/delta_rho=0.000075/0.3979=1.884896E-4 I.
Using the parameters determined above, a compensation for a variation in mass of the fluid column of the cavity of the hollow oscillating element can be performed. One exemplary way of doing this is to edit the basic equation k=(m1m0)/(1/f121/f02) to allow for the variation in the mass of the fluid column within the cavity of the hollow oscillating element. Each of the masses in this equation can be represented as a sum of the oscillating element mass and the fluid column mass. That is, mn=mnmmnF, where mnm is the oscillating element mass and mis the fluid column effective mass or xe2x80x9cactivexe2x80x9d mass. The effective mass can be determined by multiplying the density by the xe2x80x9cactivexe2x80x9d volume, which is determined as shown above. Thus, m1F=V*rho1F. The variable rho1f is a function of T1F and P1F as follows:rho1f=rhoS*PS/P1F*T1F/TS Where the subscripts s refer to density, pressure and temperature at standard or reference conditions.
Substituting rho1F into the equation for m1F, we get:m1F=V*rhoS*PS/P1F*T1F/TS Similarly, m0F=V*rhoS*PS/P0F*T0F/TS Therefore, m1xe2x88x92m=(m1m+V*rhoS*PS/P1f*T1f/TS) (m0m+V*rhoS*PS/P0F*T0F/TS)m1xe2x88x92m0=(m1mm0m)+V*rhoS*PS/TS*(T1F/P1Fxe2x88x92T0FP0F)We now substitute these mass equations into a rearranged k=(m1m0)/(1/f121/f02), as follows:(m1m0)=k (1/f121/f02)(m1mm0m)+V*rhoS*PS/TS*(T1F/P1Fxe2x88x92T0F/P0F)=k(1/f12/1f02)m1m=m0m+k(1/f1 21/f02) V*rhoS*PS/TS*(T1F/P0f/P0F)Comparing this to the original equation, we can see that the term V*rhoS*PS/TS*(T1f/P1Fxe2x88x92T0F/P0F) provides the compensation for the variation in temperature/pressure/density of the fluid column in the cavity of the hollow oscillating element. If the pressure and temperature remain unchanged, the term (T1f/P1fxe2x88x92T0f/P0f=0, and the equation reverts to the original equation of m1m=m0m+k (1/f121/f02).
But if the temperature and/or pressure vary, this term will provide an adjustment.
In the practical application of the invention, T1F, P1F, T0F, P0F, f1, f0 are observed using appropriate sensors as will be described in more detail below. The spring constant k is determined along with a known mass at conditions P0f and T0F. The resulting observed f0 allows for the calculation of k.